3.3.25 \(\int \frac {1+x^6}{x-x^7} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1593, 446, 72} \begin {gather*} \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^6)/(x - x^7),x]

[Out]

Log[x] - Log[1 - x^6]/3

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {1+x^6}{x-x^7} \, dx &=\int \frac {1+x^6}{x \left (1-x^6\right )} \, dx\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+x}{(1-x) x} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (-\frac {2}{-1+x}+\frac {1}{x}\right ) \, dx,x,x^6\right )\\ &=\log (x)-\frac {1}{3} \log \left (1-x^6\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \log (x)-\frac {1}{3} \log \left (1-x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^6)/(x - x^7),x]

[Out]

Log[x] - Log[1 - x^6]/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{x-x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x^6)/(x - x^7),x]

[Out]

IntegrateAlgebraic[(1 + x^6)/(x - x^7), x]

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fricas [A]  time = 0.39, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{3} \, \log \left (x^{6} - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/(-x^7+x),x, algorithm="fricas")

[Out]

-1/3*log(x^6 - 1) + log(x)

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giac [A]  time = 0.15, size = 16, normalized size = 1.07 \begin {gather*} \frac {1}{6} \, \log \left (x^{6}\right ) - \frac {1}{3} \, \log \left ({\left | x^{6} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/(-x^7+x),x, algorithm="giac")

[Out]

1/6*log(x^6) - 1/3*log(abs(x^6 - 1))

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maple [B]  time = 0.05, size = 36, normalized size = 2.40 \begin {gather*} \ln \relax (x )-\frac {\ln \left (x -1\right )}{3}-\frac {\ln \left (x +1\right )}{3}-\frac {\ln \left (x^{2}-x +1\right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)/(-x^7+x),x)

[Out]

-1/3*ln(x-1)-1/3*ln(x^2+x+1)-1/3*ln(x+1)+ln(x)-1/3*ln(x^2-x+1)

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maxima [B]  time = 3.01, size = 35, normalized size = 2.33 \begin {gather*} -\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{3} \, \log \left (x + 1\right ) - \frac {1}{3} \, \log \left (x - 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/(-x^7+x),x, algorithm="maxima")

[Out]

-1/3*log(x^2 + x + 1) - 1/3*log(x^2 - x + 1) - 1/3*log(x + 1) - 1/3*log(x - 1) + log(x)

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mupad [B]  time = 0.15, size = 11, normalized size = 0.73 \begin {gather*} \ln \relax (x)-\frac {\ln \left (x^6-1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 + 1)/(x - x^7),x)

[Out]

log(x) - log(x^6 - 1)/3

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sympy [A]  time = 0.12, size = 10, normalized size = 0.67 \begin {gather*} \log {\relax (x )} - \frac {\log {\left (x^{6} - 1 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)/(-x**7+x),x)

[Out]

log(x) - log(x**6 - 1)/3

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